Interfacial fluid instabilities and Kapitsa pendula

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Abstract.

The onset and development of instabilities is one of the central problems in fluid mechanics. Here we develop a connection between instabilities of free fluid interfaces and inverted pendula. When acted upon solely by the gravitational force, the inverted pendulum is unstable. This position can be stabilized by the Kapitsa phenomenon, in which high-frequency low-amplitude vertical vibrations of the base creates a fictitious force which opposes the gravitational force. By transforming the dynamical equations governing a fluid interface into an appropriate pendulum-type equation, we demonstrate how stability can be induced in fluid systems by properly tuned vibrations. We construct a “dictionary”-type relationship between various pendula and the classical Rayleigh-Taylor, Kelvin-Helmholtz, Rayleigh-Plateau and the self-gravitational instabilities. This makes several results in control theory and dynamical systems directly applicable to the study of tunable fluid instabilities, where the critical wavelength depends on the external forces or the instability is suppressed entirely. We suggest some applications and instances of the effect ranging in scale from microns to the radius of a galaxy.

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Keywords

Flowing Matter: Liquids and Complex Fluids 

References

  1. 1.
    M.C. Cross, P.C. Hohenberg, Rev. Mod. Phys. 65, 851 (1993)ADSCrossRefGoogle Scholar
  2. 2.
    S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Dover Publications, Inc., Mineola, NY, 1981). Google Scholar
  3. 3.
    P.G. Drazin, Introduction to Hydrodynamic Stability (Cambridge University Press, Cambridge, UK, 2002)Google Scholar
  4. 4.
    H.J. Kull, Phys. Rep. 206, 197 (1991)ADSCrossRefGoogle Scholar
  5. 5.
    K. Baldwin, M. Scase, R. Hill, Nat. Sci. Rep. 5, 11706 (2015)ADSCrossRefGoogle Scholar
  6. 6.
    A. Poehlmann, R. Richter, I. Rehberg, J. Fluid Mech. 732, 1 (2013)CrossRefGoogle Scholar
  7. 7.
    X. Chen, F. Eliot, J. Fluid Mech. 560, 395 (2006)ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Mathew J. Russo, Paul H. Steen, Phys. Fluids A 1, 1926 (1989)ADSCrossRefGoogle Scholar
  9. 9.
    M. Marr-Lyon, D. Thiessen, P. Marston, J. Fluid Mech. 351, 345 (1997)ADSCrossRefGoogle Scholar
  10. 10.
    M. Marr-Lyon, D. Thiessen, P. Marston, Phys. Rev. Lett. 86, 2293 (2001)ADSCrossRefGoogle Scholar
  11. 11.
    N. Bertin, R. Wunenburger, E. Brasselet, J.-P. Delville, Phys. Rev. Lett. 105, 164501 (2010)ADSCrossRefGoogle Scholar
  12. 12.
    D.V. Lyubimov, A.A. Cherepanov, Izv. Akad. Nauk SSSR, Mekh. Zhidk. Gaza 6, 3 (1991)ADSGoogle Scholar
  13. 13.
    G. Gandikota, D. Chatain, T. Lyubimova, D. Beysens, Phys. Rev. E 89, 063003 (2014)ADSCrossRefGoogle Scholar
  14. 14.
    V. Shevtsova, Y.A. Gaponenko, V. Yasnou, A. Mialdun, A. Nepomnyashchy, J. Fluid Mech. 795, 409 (2016)ADSCrossRefGoogle Scholar
  15. 15.
    V.I. Arnold, Mathematical Methods of Classical Mechanics (Springer, New York, NY, 2010)Google Scholar
  16. 16.
    A. Stephenson, Mem. Proc. Manchester Lit. Philos. Soc. 52, 1 (1908)Google Scholar
  17. 17.
    A. Stephenson, Philos. Mag. 17, 765 (1909)CrossRefGoogle Scholar
  18. 18.
    P.L. Kapitsa, Dynamical stability of a pendulum when its point of suspension vibrates, Collected Papers of P.L. Kapitsa, Vol. II (Pergamon Press, 1965) pp. 714--725Google Scholar
  19. 19.
    P.L. Kapitsa, Pendulum with a vibrating suspension, Collected Papers of P.L. Kapitsa, Vol. II (Pergamon Press, 1965) pp. 726--737Google Scholar
  20. 20.
    V.N. Chelomei, Dokl. Akad. Nauk SSSR 110, 345 (1983)Google Scholar
  21. 21.
    M. Levi, SIAM Rev. 30, 639 (1988)MathSciNetCrossRefGoogle Scholar
  22. 22.
    M. Levi, H. Broer, Arch. Ration. Mech. Anal. 131, 225 (1995)CrossRefGoogle Scholar
  23. 23.
    M. Levi, Int. J. Bifurcat. Chaos 15, 2747 (2005)CrossRefGoogle Scholar
  24. 24.
    W. Paul, Rev. Mod. Phys. 62, 531 (1990)ADSCrossRefGoogle Scholar
  25. 25.
    J. Holyst, W. Wojciechowski, Physica A 324, 388 (2003)ADSMathSciNetCrossRefGoogle Scholar
  26. 26.
    T.B. Benjamin, F. Ursell, Proc. R. Soc. Lond. A 225, 505 (1954)ADSCrossRefGoogle Scholar
  27. 27.
    F. Troyon, R. Gruber, Phys. Fluids 14, 2069 (1971)ADSCrossRefGoogle Scholar
  28. 28.
    G.H. Wolf, Z. Phys. 227, 291 (1969)ADSCrossRefGoogle Scholar
  29. 29.
    G.H. Wolf, Phys. Rev. Lett. 24, 444 (1970)ADSCrossRefGoogle Scholar
  30. 30.
    Inga Koszalka, Vibrating pendulum and stratified fluids, in Geophysical Fluid Dynamics Proceedings Volumes (WHOI, 2005)Google Scholar
  31. 31.
    A. Weathers, B. Folie, B. Liu, S. Childress, J. Zhang, J. Fluid Mech. 650, 415 (2010)ADSCrossRefGoogle Scholar
  32. 32.
    L.D. Landau, E.M. Lifshitz, Mechanics, 3rd edition (Pergamon Press, Oxford, 1986)Google Scholar
  33. 33.
    A. Seyranian, A. Mailybaev, Multiparameter Stability Theory with Mechanical Applications (World Scientific, NJ, 2004)Google Scholar
  34. 34.
    A. Mailybaev, A. Seyranian, J. Sound Vibrat. 323, 1016 (2009)ADSCrossRefGoogle Scholar
  35. 35.
    H. Broer, I. Hoveijn, M. van Noort, C. Simó, G. Vegter, J. Dyn. Differ. Equ. 16, 897 (2004)CrossRefGoogle Scholar
  36. 36.
    H.W. Broer, I. Hoveijn, M. van Noort, G. Vegter, J. Differ. Equ. 157, 120 (1999)ADSCrossRefGoogle Scholar
  37. 37.
    J. Wesson, Phys. Fluids 13, 762 (1970)ADSCrossRefGoogle Scholar
  38. 38.
    K. Kumar, L. Tuckerman, J. Fluid Mech. 279, 49 (1994)ADSMathSciNetCrossRefGoogle Scholar
  39. 39.
    D. Horsley, L. Forbes, J. Eng. Math. 79, 13 (2013)CrossRefGoogle Scholar
  40. 40.
    P. Chen, Z. Luo, S. Güven, S. Tasoglu, D.V. Ganesan, A. Weng, U. Demirci, Adv. Mater. 26, 5936 (2014)CrossRefGoogle Scholar
  41. 41.
    R. Krechetnikov, J.E. Marsden, Physica D 214, 25 (2006)ADSMathSciNetCrossRefGoogle Scholar
  42. 42.
    D.J. Acheson, T. Mullin, Nat. Corresp. 366, 215 (1993)Google Scholar
  43. 43.
    D.J. Acheson, Proc. R. Soc. Lond. A 443, 239 (1993)ADSCrossRefGoogle Scholar
  44. 44.
    T. Mullin, A. Champneys, B. Fraser, J. Galan, D. Acheson, Proc. R. Soc. Lond. A 459, 539 (2003)ADSCrossRefGoogle Scholar
  45. 45.
    B. Fraser, A. Champneys, Proc. R. Soc. Lond. A 458, 1353 (2002)ADSCrossRefGoogle Scholar
  46. 46.
    M.-R. Alam, Y. Liu, D.K.P. Yue, J. Fluid Mech. 624, 191 (2009)ADSMathSciNetCrossRefGoogle Scholar
  47. 47.
    H. Jeffreys, Proc. R. Soc. Lond. A 107, 189 (1925)ADSCrossRefGoogle Scholar
  48. 48.
    A. Jenkins, Phys. Rep. 525, 167 (2013)ADSMathSciNetCrossRefGoogle Scholar
  49. 49.
    M. Ruijgrok, A. Tondl, F. Verhulst, ZAMM 73, 255 (1993)ADSCrossRefGoogle Scholar
  50. 50.
    A. Bloch, P. Hagerty, A.G. Rojo, M.I. Weinstein, J. Stat. Phys. 115, 1073 (2004)ADSCrossRefGoogle Scholar
  51. 51.
    G.K. Batchelor, An Introduction to Fluid Mechanics, 3rd edition (Cambridge University Press, Cambridge, 1976)Google Scholar
  52. 52.
    D. Merkt, A. Pototsky, M. Bestehorn, U. Thiele, Phys. Fluids 17, 064104 (2005)ADSMathSciNetCrossRefGoogle Scholar
  53. 53.
    W.Y. Jiang, S.P. Lin, Phys. Fluids 17, 054105 (2005)ADSCrossRefGoogle Scholar
  54. 54.
    J. Atencia, D. Beebe, Nature 437, 648 (2005)ADSCrossRefGoogle Scholar
  55. 55.
    P. Trinh, H. Kim, N. Hammoud, P. Howell, S.J. Chapman, H. Stone, Phys. Fluids 26, 051704 (2014)ADSCrossRefGoogle Scholar
  56. 56.
    E.R. Harrison, Phys. Rev. D 1, 2726 (1970)ADSCrossRefGoogle Scholar
  57. 57.
    Ya.B. Zel’dovich, Mon. Not. R. Astron. Soc. 160, 1P (1972)ADSCrossRefGoogle Scholar
  58. 58.
    G.F. Smoot et al., Astrophys. J. 396, L1 (1992)ADSCrossRefGoogle Scholar
  59. 59.
    Carl H. Gibson, Primordial viscosity, diffusivity, Reynolds number and turbulence in the beginnings of gravitational structure formation, PhD Thesis, UCSD, San Diego, CA, 1996Google Scholar
  60. 60.
    K. Subramanian, The origin, evolution and signatures of primordial magnetic fields, arXiv:1504.02311v1 (2015)

Copyright information

© EDP Sciences, SIF, Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.School of EngineeringBrown UniversityProvidenceUSA
  2. 2.Program for Evolutionary DynamicsHarvard UniversityCambridgeUSA

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